Integration formulas by parts is a technique used to integrate the product of two functions. It is based on the product rule of differentiation and involves breaking the integrand into two parts and integrating one part while differentiating the other.
The formula for integration by parts is ∫u dv = uv – ∫v du, where u and v are functions, and dv and du are their differentials. This method is often used to integrate functions involving products of algebraic, logarithmic, exponential, trigonometric, and inverse trigonometric functions.
In general, Integration is a mathematical operation that involves finding the integral of a function. It is the process of determining the area under a curve and has many applications in calculus, physics, and engineering. The result of integration is called the antiderivative or indefinite integral of a function.
Integration formulas chart & Integration formulas PDF class 12
The integration formulas and also Integration formulas by parts are presented generally in the sets of formulas that follow. A more complicated set of integration formulae are also presented, along with fundamental integration formulas, integration of trigonometric ratios, inverse trigonometric functions, the product of functions, and more.
In essence, integration is a process of connecting the pieces to form the whole. It is the reverse of differentiation in action. Accordingly, the basic integration equation is f'(x) dx = f(x) + C. This generates the following integration formulas given below.
Integration formulas for class 12- Introduction and Integration formulas list
Integration formulas for class 12 are mathematical expressions that provide a shortcut for finding antiderivatives or indefinite integrals of commonly occurring functions. These Integration formulas for class 12 can be used to evaluate integrals without having to go through the entire process of integration.
Some examples of integration formulas include the power rule, which states that the integral of x^n is (x^(n+1))/(n+1), and the trigonometric identities such as the integral of sin(x) which is -cos(x) + C. There are many other integration formulas for different types of functions, including exponential, logarithmic, hyperbolic, and inverse trigonometric functions.
Integration formulas basics Class 12 CBSE Complete List:
∫ 1 dx = x + C
∫ a dx = ax+ C
∫ xn dx = ((xn+1)/(n+1))+C ; n≠1
∫ sin x dx = – cos x + C
∫ cos x dx = sin x + C
∫ sec2x dx = tan x + C
∫ cosec2x dx = -cot x + C
∫ sec x (tan x) dx = sec x + C
∫ cosec x ( cot x) dx = – cosec x + C
∫ (1/x) dx = ln |x| + C
∫ ex dx = ex+ C
∫ ax dx = (ax/ln a) + C ; a>0, a≠1
∫ tanx.dx =log|secx| + C
∫ cotx.dx = log|sinx| + C
∫ secx.dx = log|secx + tanx| + C
∫ cosecx.dx = log|cosecx – cotx| + C
Integration formulas for trigonometric functions Class 12 CBSE Complete List:
∫ 1/(1 +x2).dx = -cot-1x + C
∫ 1/x√(x2- 1).dx = sec-1x + C
∫ 1/x√(x2- 1).dx = -cosec-1x + C
∫ 1/√(1 – x2).dx = sin-1x + Cfor
∫ 1/(1 – x2).dx = -cos-1x + C
∫ 1/(1 + x2).dx = tan-1x + C
Integration formulas list (Difficult) for Class 12 CBSE:
∫ √(x2 + a2).dx =1/2.x.√(x2 + a2)+ a2/2 . log|x + √(x2 + a2)| + C
∫ 1/(x2 + a2).dx = 1/a.tan-1x/a + C
∫ 1/√(x2- a2)dx = log|x +√(x2- a2)| + C
∫ √(x2- a2).dx =1/2.x.√(x2- a2)-a2/2 log|x + √(x2- a2)| + C
∫ 1/√(a2- x2).dx = sin-1x/a + C
∫ 1/(x2- a2).dx = 1/2a.log|(x – a)(x + a| + C
∫ 1/(a2- x2).dx =1/2a.log|(a + x)(a – x)| + C
∫ 1/√(x2 + a2).dx = log|x + √(x2 + a2)| + C
∫ √(a2- x2).dx = 1/2.x.√(a2- x2).dx + a2/2.sin-1 x/a + C
Integration formulas by parts – Advanced Formula and Explanation
Integration formulas by parts is a technique used to find the antiderivative of a product of two functions. It is based on the product rule of differentiation, which states that the derivative of a product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function.
The Integration formulas by parts are ∫u dv = uv – ∫v du, where u and v are functions, and dv and du are their differentials.
To use these Integration formulas by parts, we select u and dv in such a way that the integral on the right-hand side is easier to evaluate than the original integral on the left-hand side.
We then apply the formula and repeat the process until the integral can be evaluated.
For example, let’s consider the integral of x*e^x. We can choose u = x and dv = e^x dx, which gives us du = dx and v = e^x. Applying the integration by parts formula, we get:
∫xe^x dx = xe^x – ∫e^x dx
= x*e^x – e^x + C
where C is the constant of integration. This is the antiderivative of x*e^x.
Integration by parts that is Integration formulas by parts is a powerful tool for evaluating a wide variety of integrals. It is often used to integrate functions involving products of algebraic, logarithmic, exponential, trigonometric, and inverse trigonometric functions. For more examples see here.
You might like reading: For formulas of Surface Area and Volume class 10 & Competitive Exams
Integration formulas class 12 – Conclusion
Lastly, in simple terms integration is the process of finding an integral. Areas, volumes, displacement, and other important quantities can all be calculated using integrals in mathematics. Typically, we refer to definite integrals when we discuss integrals.
Indefinite integrals are used for ant derivatives. One of the two main calculus topics in mathematics that evaluates the rate of change of any function with respect to its variables, along with differentiation, is integration. It’s a vast subject that is addressed in classes at the highest levels, like Classes 11 and 12.
You might like reading this: Syllabus of BARC for Computer Science 2023 [Latest]
Integration formulas by parts FAQs
What are the Integration formulas by parts?
The Integration formulas by parts are ∫u dv = uv – ∫v du, where u and v are functions, and dv and du are their differentials.
How do I choose u and dv in Integration formulas by parts?
In general in Integration formulas by parts, you should be chosen to be part of the integrand that becomes simpler when differentiated, and dv should be chosen to be the part that becomes simpler when integrated. However, the choice of u and dv can depend on the specific problem at hand.
How many times should I apply integration by parts?
You should repeat the Integration formulas by parts process as many times as necessary to obtain an integral that can be evaluated.
How do I know when to use Integration formulas by parts?
Integration formulas by parts are useful when the integrand is a product of two functions, and neither function can be easily integrated by itself.
Are there any tricks to make Integration formulas by parts easier?
Sometimes, it can be helpful to choose u and dv in such a way that the resulting integral on the right-hand side is simpler than the original integral. It can also be helpful to look for patterns in the integrand and choose u and dv accordingly.
